Anyone but Him: The Complexity of Precluding an Alternative∗
Edith Hemaspaandra
Lane A. Hemaspaandra
Jörg Rothe
Department of Computer Science
Rochester Institute of Technology
Rochester, NY 14623, USA
eh@cs.rit.edu
Department of Computer Science
University of Rochester
Rochester, NY 14627, USA
lane@cs.rochester.edu
Institut für Informatik
Heinrich-Heine-Universität Düsseldorf
40225 Düsseldorf, Germany
rothe@cs.uni-duesseldorf.de
Abstract
Preference aggregation in a multiagent setting is a central
issue in both human and computer contexts. In this paper,
we study in terms of complexity the vulnerability of preference aggregation to destructive control. That is, we study the
ability of an election’s chair to, through such mechanisms as
voter/candidate addition/suppression/partition, ensure that a
particular candidate (equivalently, alternative) does not win.
And we study the extent to which election systems can make
it impossible, or computationally costly (NP-complete), for
the chair to execute such control. Among the systems we
study—plurality, Condorcet, and approval voting—we find
cases where systems immune or computationally resistant to
a chair choosing the winner nonetheless are vulnerable to the
chair blocking a victory. Beyond that, we see that among
our studied systems no one system offers the best protection
against destructive control. Rather, the choice of a preference aggregation system will depend closely on which types
of control one wishes to be protected against. We also find
concrete cases where the complexity of or susceptibility to
control varies dramatically based on the choice among natural tie-handling rules.
Key words: preferences, computational complexity, multiagent systems.
Introduction
Voting systems provide a broad model for aggregating preferences in a multiagent setting. The literature on voting is
vast and active, and spans such areas as AI, complexity, economics, operations research, and political science. As noted
by Conitzer, Lang, and Sandholm (2003), voting has been
proposed as a mechanism for use in decision-making in various computational settings, including planning (Ephrati &
Rosenschein 1991; 1993) and collaborative filtering (Pennock, Horvitz, & Giles 2000). Voting also may be useful in
many large-scale computer settings. Examples of much recent interest include the (web-page) rank aggregation problem, and related issues of reducing “spam” results in web
∗
Supported in part by the NSF under grants NSF-CCR-0311021
and NSF-CCF-0426761 and by the DFG under grant RO 1202/9-1.
This work was done in part while the authors were visiting Klaus
Wagner at the University of Würzburg, and while the first author
was on sabbatical at the University of Rochester.
c 2005, American Association for Artificial IntelliCopyright gence (www.aaai.org). All rights reserved.
search and improving similarity search, for which the use
of voting systems has been proposed (Dwork et al. 2001;
Fagin, Kumar, & Sivakumar 2003). In such an automated
setting, it is natural to imagine decisions with thousands or
millions of “voters” and “candidates.”
In the seminal paper “How hard is it to control an election?” (Bartholdi, Tovey, & Trick 1992), the issue of constructive control of election systems is studied: How hard
is it for a chair (who knows all voters’ preferences) to—
through control of the voter or candidate set or of the
partition structure of an election—cause a given candidate
(equivalently, alternative) to be the (unique) winner?1 They
studied plurality and Condorcet voting, and seven natural types of control: adding candidates, suppressing candidates, partition of candidates, run-off partition of candidates,
adding voters, suppressing voters, and partition of voters.
They found that in some cases there is immunity to constructive control (if his/her candidate was not already the2
unique winner, no action of the specified type by the chair
can make the candidate the unique winner), in some cases
there is (computational) resistance to constructive control
(it is NP-complete to decide whether the chair can achieve
his/her desired outcome), and in some cases the system is
(computationally) vulnerable to constructive control (there
is a polynomial-time algorithm that will tell the chair how to
achieve the desired outcome whenever possible3 ).
In this paper, we obtain results for each of their 14 cases
(two preference aggregation systems, each under seven con1
In their model, which is also adopted here, the chair has complete information on the voters’ preferences. This is a natural assumption in many situations. For example, in a computer science
department, after endless discussions, most people know what each
person’s position is on key issues. Also, since the case where complete information is available to the chair is a special subcase of the
more general setting that allows information to be specified with
any level of completeness, lower bounds obtained in the complete
information setting are inherited by any natural incomplete information model.
2
Really “a unique winner,” since there may be no winner at all,
but we’ll usually write “the unique winner” when this is clear from
context.
3
This is more like “computationally certifiably-vulnerable,” see
Definition 1. Vulnerability as defined in (Bartholdi, Tovey, & Trick
1992) means one can quickly decide if there exists a way for the
chair to achieve the desired outcome.
AAAI-05 / 95
trol schemes) in the setting of destructive control. In contrast
with constructive control, in which a chair tries to ensure that
a specified desirable candidate is the (unique) winner, in destructive control the chair tries to ensure that a specified detested candidate is not the (unique) winner. Regarding the
naturalness of destructivity, the light-hearted title of this paper tries to reflect the fact that, in human terms, one often
hears feelings expressed that focus strategically on precluding one candidate, and of course in other settings this also
may be a goal. Regarding the reality of electoral control,
from targeted “get-out-the-vote” advertisements of parties
and candidates to (alleged) voter suppression efforts by independent groups, from the way a committee chair groups alternatives to any case where a faculty member hands out student course evaluations on a day some malcontent students
are not in class, it is hard to doubt that the desire for electoral
control—both destructive and constructive—is a real one.
Destruction has been previously studied by Conitzer,
Lang, and Sandholm (2002; 2003), but in the setting of
election manipulation—in which some voter(s) knowing all
other voters’ preferences are free to shift their preferences
dynamically to affect the outcome. In contrast, in this paper
we study destruction in the very different setting of electoral
control (Bartholdi, Tovey, & Trick 1992)—where a chair,
given fixed and unchangeable voter preferences, tries to influence the outcome via procedural/access means.
One might ask, “Why bother studying destructive control, since any rational chair would prefer to assert constructive control?” The answer is that it is plausible—and our
results show it is indeed the case—that destructive control
may be possible in settings in which constructive control is
not. Informally put, destructive control may be easier for
the chair to assert. For example, we prove formally that of
the seven types of constructive control of Condorcet elections that Bartholdi, Tovey, and Trick (1992) study, the four
they showed not vulnerable to constructive control are all
vulnerable to destructive control. The remaining three cases
regarding Condorcet voting are vulnerable to constructive
control (Bartholdi, Tovey, & Trick 1992), but we show that
they are immune to destructive control.4
4
Savvy readers may wonder whether the last statement is impossible. After all, to ensure that the despised candidate c is not
the unique winner, we simply have to ask whether at least one
of the other candidates can be ensured to unique-win-or-tie-forwinner. Formally put, destructive control polynomial-time disjunctively truth-table reduces (Ladner, Lynch, & Selman 1975) to
constructive control (redefined to embrace ties), and so the destructive control problem can (within a polynomial factor) be no
less hard computationally than the (redefined) constructive control
problem (this is noted in a different setting by Conitzer and Sandholm (2002)), seemingly (though not really) contrary to the three
cases just mentioned. The brief explanation of why this does not
cause a paradox lies in the word “computational”: Although immunity is the most desirable case in terms of security from control, the complexity of recognizing whether a given candidate can
be precluded from winning in immune cases will most typically
be in P—after all, we can never, when immunity holds, change
a given candidate from unique winner to not the unique winner,
so the related decision problem is typically easy. (Technical side
remark: We say “will most typically be in P/is typically” rather
Table 1 summarizes our results on the complexity of destructively controlling Condorcet, plurality, and approval
elections. We also when needed obtain, for comparative purposes, new results on the complexity of constructive control,
and Table 1 displays those and also constructive control results of Bartholdi, Tovey, and Trick (1992). All entries in
boldface in Table 1 are new results obtained in this paper; the
other results are due to Bartholdi, Tovey, and Trick (1992).
For each boldface “V” in the table, “certifiably-vulnerable”
is in fact also achieved by our theorems. We mention in
passing that for nonboldface “V”s in the table, “certifiablyvulnerable” can be seen directly from or by modifying the
algorithms of Bartholdi, Tovey, and Trick (1992).
For control-by-partition problems—which will involve
subelection(s)—we distinguish between the models TiesEliminate (TE, for short) and Ties-Promote (TP, for short),
which define what happens when there are ties in a subelection (before the final election), namely, all participating candidates are eliminated (TE), or all who tie for winner move
forward (TP). Note that these models do not apply to Condorcet voting, under which when a winner exists s/he is inherently unique; so the TE/TP distinction is made only for
plurality and approval voting.
The natural conclusion to draw from our results is that
when selecting an election/preference aggregation system,
one should at least be aware of the issue of the system’s
vulnerability to control—and, beyond that, one’s choice of
system will depend closely on which types of immunity or
computational resistance one most values. Our results also
show that constructive and destructive control often differ
greatly: A system immune to constructive control may be
vulnerable to destructive control, and vice versa. Finally,
our results show that—in contrast with some comments in
earlier papers—breaking ties is far from a minor issue: For
both voting types where tie-handling rules are meaningful,
we find cases where the complexity of or susceptibility to
control varies dramatically based on the choice among natural tie-handling rules.
Preliminaries
We first define the three voting systems considered. In approval voting, each voter votes “Yes” or “No” for each candidate. (So, for approval voting, a voter’s preferences are
reflected by a 0-1 vector.) All candidates with the maximum
number of “Yes” votes are winners.
Plurality and Condorcet voting are defined in terms of
strict preferences. For them, an election is given by a preference profile, a pair (C, V ) such that C is a set of candidates
and V is the multiset (henceforth, we’ll just say set, as a
shorthand) of the voters’ preference orders on C. We assume
that the preference orders are irreflexive and antisymmetric
than “will be in P/is” because for impractical systems that—unlike
those here—have winner-testing problems that are not in P, it is in
concept possible that one can have immunity and yet also have the
related language problem not belong to P.)
The disjunctive-truth-table connection mentioned above explains why, if P 6= NP, it is impossible to have computational
resistance to destructive control hold for any problem that, when
redefined to embrace ties, is vulnerable to constructive control.
AAAI-05 / 96
Control by
Adding Candidates
Deleting Candidates
Partition
of Candidates
Run-off Partition
of Candidates
Adding Voters
Deleting Voters
Partition
of Voters
Plurality
Constructive Destructive
R
R
R
R
TE: R
TE: R
TP: R
TP: R
TE: R
TE: R
TP: R
TP: R
V
V
V
V
TE: V
TE: V
TP: R
TP: R
Condorcet
Constructive Destructive
I
V
V
I
V
I
V
I
R
R
R
V
V
V
Approval
Constructive Destructive
I
V
V
I
TE: V
TE: I
TP: I
TP: I
TE: V
TE: I
TP: I
TP: I
R
V
R
V
TE: R
TE: V
TP: R
TP: V
Table 1: Summary of results. Results new to this paper are in boldface. Nonboldface results are due to Bartholdi, Tovey, and
Trick (1992). Key: I = immune, R = resistant, V = vulnerable, TE = Ties-Eliminate, TP = Ties-Promote.
(i.e., every voter has strict preferences over the candidates),
complete (i.e., every voter ranks each candidate), and transitive.
A voting system is a rule for how to determine the winner(s) of an election. Formally, any voting system is defined
to be a (social choice) function mapping any given preference profile (or the analog with voters’ 0-1 vectors for the
approval voting case) to society’s aggregate choice set, the
set of candidates who have won the election.
In plurality voting, each candidate with a maximum number of “first preference among the candidates in the election” voters for him/her wins. In Condorcet voting, for each
c ∈ C, c is a winner if and only if for each d ∈ C with
d 6= c, c defeats d by a strict majority of votes in a pairwise
election between them based on the voters’ preferences.
The Condorcet Paradox observes that whenever there are
at least three candidates, due to cyclic aggregate preference rankings Condorcet winners may not exist (Condorcet
1785). That is, the set of winners may be empty. However, a Condorcet winner is unique whenever one does exist. In the case of plurality and approval voting, due to
ties, there may exist multiple winners. Regarding ties, we—
following Bartholdi, Tovey, and Trick (1992) to best allow
comparison—focus in our control problems on creating a
unique winner (constructive), and precluding a candidate
from being the unique winner (destructive). (Ties in subelections, for the partition problems, are handled via the TE
and TP rules described earlier.)
Results
The issue of control of an election by the authority conducting it (called the chair) can be studied under a variety of
models and scenarios. For plurality and Condorcet voting,
Bartholdi, Tovey, and Trick (1992)—which for the rest of
this section will be referred to as “BTT92”—study constructive control by adding candidates, deleting candidates, partition of candidates, run-off partition of candidates, adding
voters, deleting voters, and partition of voters. In their setting, the chair’s goal is to make a given candidate uniquely
win the election. Analogously, we consider in turn the corresponding seven destructive control problems, where the
chair’s goal is to preclude a given candidate from being the
unique winner. For each of these seven we define the problem and present prior results and our results. (Formally,
for each type of control one defines a decision problem and
studies its computational complexity.) To make comparisons
as easy as possible, we in stating these seven control problems whenever possible exactly follow BTT92’s wording—
modified to the destructive case (and when we diverge, we
explain why and how).
Destructive Control by Adding Candidates As is common, we state our decision problems as “Given” instances,
and a related Yes/No question. The language in each case is
the set of all instances for which the answer is Yes.
Given: A set C of qualified candidates and a distinguished
candidate c ∈ C, a set D of possible spoiler candidates,
and a set V of voters with preferences (in the approval
case, the “preferences” will, as always for that case, actually be 0-1 vectors) over C ∪ D.
Question: Is there a choice of candidates from D whose
entry into the election would assure that c is not the unique
winner?
The above type of destructive control captures the idea
that the chair tries to dethrone the despised candidate c by
introducing new “spoiler” candidates.
With this first problem stated, now is a good time to define
our notions of control. Our terminology will closely follow
the notions in BTT92, to allow comparison.
Definition 1 We say that a voting system is immune to destructive control in a given model (of control) if it is never
possible for the chair to change a given candidate from
the unique winner to being not the unique winner by using
his/her allowed model of control. If a system is not immune
to a type of control, it is said to be susceptible to that type of
control.
A voting system is said to be (computationally) vulnerable
to control if it is susceptible to control and the corresponding
language problem is computationally easy (i.e., solvable in
polynomial time). If a system is not just vulnerable but one
can, given a control problem, produce in polynomial time
AAAI-05 / 97
the actual action of the chair to execute control the “best”
way (namely, by adding or deleting the smallest number of
candidates or voters for add/delete problems; for partition
problems, any legal partition that works is acceptable), we
say the system is (computationally) certifiably-vulnerable to
control.5
A voting system is said to be resistant to control if it is susceptible to control but the corresponding language problem
is computationally hard (i.e., NP-complete).6
For the theory of NP-completeness, see, e.g., (Garey &
Johnson 1979; Hopcroft & Ullman 1979). For space reasons we do not explicitly define the corresponding notions
for constructive control (except briefly in the introduction)
and the analogous seven constructive control decision problems from BTT92, but except when noted below they are
exactly analogous. For example, for the above control scenario, the analogous constructive problem pairs the same
“Given” with the question “Is there a choice of candidates
from D whose entry into the election would assure that c is
the unique winner?”
As to what is known about Control by Adding Candidates,
BTT92 showed that plurality is resistant and Condorcet is
immune. Our results are:
Theorem 2 Approval (voting) is immune to constructive control by adding candidates, and plurality, Condorcet, and approval (voting) are respectively resistant,
vulnerable/certifiably-vulnerable, and vulnerable/certifiably-vulnerable to destructive control by adding candidates.
So, though Condorcet and approval are immune to constructive control of this sort, they both are vulnerable to destructive control. This reverses itself for:
Destructive Control by Deleting Candidates
Given: A set C of candidates, a distinguished candidate c ∈
C, a set V of voters, and a positive integer k < ||C||.
Question: Are there k or fewer candidates other than c in
C whose disqualification would assure that c is not the
unique winner?
5
For the seven problems studied here, certifiably-vulnerable implies vulnerable (but we list both, since if one studied add/delete
problems stated not in terms of “is there some subset” or “by
adding/deleting ≤ k” but rather in terms of “by adding/deleting
exactly k,” then for certain systems the implication need not hold).
6
It would be more natural to define resistance as meaning the
corresponding language is (many-one) NP-hard. However, in this
paper, we define resistance in terms of NP-completeness. One reason is that this matches the way the term is used by BTT92. More
importantly, all the problems discussed in this paper have obvious
NP upper bounds since testing whether a given candidate has won a
given election for the systems considered here is obviously in P. So
for the problems in this paper, NP-completeness and NP-hardness
stand or fall together. We mention in passing that there are natural election systems whose complexity seems beyond NP. The first
such case established was for the election system defined by Lewis
Carroll in 1876 (Dodgson 1876), where even the complexity of determining whether a given candidate has won is now known to be
hard for parallel access to NP (Hemaspaandra, Hemaspaandra, &
Rothe 1997).
In this type of control, the chair seeks to influence the
outcome of the election by suppressing certain candidates
(other than c), in hopes that their voters now support another
candidate to ensure stopping c. Note that this formalization
is not a perfect analog of the constructive case of BTT92
in that we explicitly prevent deleting c, since otherwise any
voting system in which the winners can efficiently be determined would be trivially vulnerable to this type of control.
Here, BTT92 establish for constructive control resistance
for plurality and immunity for Condorcet. Our results are:
Theorem 3 Approval is vulnerable/certifiably-vulnerable
to constructive control by deleting candidates. Plurality,
Condorcet, and approval voting are respectively resistant,
immune, and immune to destructive control by deleting candidates.
We now handle jointly the two types of partition of candidates, since they yield identical results.
Destructive Control by Partition of Candidates
Given: A set C of candidates, a distinguished candidate c ∈
C, and a set V of voters.
Question: Is there a partition of C into C1 and C2 such
that c is not the unique winner in the sequential two-stage
election in which the winners in the subelection (C1 , V )
who survive the tie-handling rule move forward to face
the candidates in C2 (with voter set V )?
Destructive Control by Run-Off Partition of Candidates
Given: A set C of candidates, a distinguished candidate c ∈
C, and a set V of voters.
Question: Is there a partition of C into C1 and C2 such that
c is not the unique winner of the election in which those
candidates surviving (with respect to the tie-handling
rule) subelections (C1 , V ) and (C2 , V ) have a run-off
with voter set V .
These two types of control express settings in which the
chair tries to, overall, partition the candidates in such a
clever way that the hated candidate c fails to be the unique
winner—one via a cascading setup, and one via a run-off
setup. Here, BTT92 show that for constructive control plurality is resistant (and their results hold in both our TE and
TP models) and Condorcet is vulnerable. Our results are:
Theorem 4 Approval is vulnerable/certifiably-vulnerable
to constructive control by partition of candidates and runoff partition of candidates in model TE and immune to constructive control by partition of candidates and run-off partition of candidates in model TP. Plurality, Condorcet, and
approval voting are, in models TE and TP, respectively resistant, immune, and immune to destructive control by partition
of candidates and by run-off partition of candidates.
So Condorcet, though vulnerable to constructive control, is
immune to destructive control here. And, perhaps more interesting, for constructive control, approval changes from
vulnerable to immune depending on the tie-handling rule.
We now turn to control of the voter set. The intuition behind seeking destructive control by adding or deleting voters
is clear, e.g., getting out the vote and vote suppression. We
handle these two cases together as their results are identical.
AAAI-05 / 98
Destructive Control by Adding Voters
Given: A set of candidates C and a distinguished candidate
c ∈ C, a set V of registered voters, an additional set W of
yet unregistered voters (both V and W have preferences
over C), and a positive integer k ≤ ||W ||.
Question: Are there k or fewer voters from W whose registration would assure that c is not the unique winner?
Destructive Control by Deleting Voters
Given: A set of candidates C, a distinguished candidate c ∈
C, a set V of voters, and a positive integer k ≤ ||V ||.
Question: Are there k or fewer voters in V whose disenfranchisement would assure that c is not the unique winner?
Here, BTT92 show that for constructive control plurality
is vulnerable and Condorcet is resistant. Our results are:
Theorem 5 Approval is resistant to constructive control
by adding voters and by deleting voters. Plurality, Condorcet, and approval voting are all vulnerable/certifiablyvulnerable to destructive control by adding voters and by
deleting voters.
So Condorcet and approval, though resistant to constructive
control, are vulnerable to destructive control here.
The final problem here results in a surprise.
Destructive Control by Partition of Voters
Given: A set of candidates C, a distinguished candidate c ∈
C, and a set V of voters.
Question: Is there a partition of V into V1 and V2 such that
c is not the unique winner in the hierarchical two-stage
election in which the survivors of (C, V1) and (C, V2) run
against each other with voter set V ?
In this last type of control, the voter set is partitioned into
two “subcommittees” that both separately select their “nominees,” who run against each other in the final decision stage.
Unlike BTT92, we again distinguish between the two models Ties-Eliminate and Ties-Promote defined above. That is,
in the Ties-Eliminate model, if two or more candidates tie
for winning in a subcommittee’s election, no candidate is
nominated by that subcommittee. In contrast, in the TiesPromote model, all the candidates who tie for winning in
a subcommittee’s election are nominated to run in the final
decision stage. If both subcommittees nominate the same
candidate (and no one else), we by convention declare this
candidate the run-off winner—one cannot eliminate oneself.
We mention that both of our two tie-handling models, TE
and TP, differ from the model adopted in BTT92, where they
for vulnerability results about this problem adopt a third
model in which ties are handled not by a tie-handling rule
but rather by changing the decision problem itself to require
the chair to find a partition that completely avoids ties in any
subcommittee. We find our model the more natural, but we
mention that they obtained for this case, in their tie model,
a constructive-control vulnerability result for plurality. For
Condorcet and constructive control, they proved that resistance holds. Our results are:
Theorem 6 Approval is resistant to constructive control by partition of voters in models TE and TP,
vulnerable/certifiably-vulnerable to destructive control by
partition of voters in models TE and TP. Plurality is
vulnerable/certifiably-vulnerable to both constructive and
destructive control by partition of voters in model TE,
and is resistant to both constructive and destructive control by partition of voters in model TP. Condorcet is
vulnerable/certifiably-vulnerable to destructive control by
partition of voters.
The most striking behavior here is that plurality voting varies
between being vulnerable and being resistant, depending on
the tie-handling rule. The loose intuition for this is that in
TE, at most one candidate wins each subcommittee and in
polynomial time we can explore every way this can happen.
In contrast, under TP potentially any subset of candidates
may move forward, and in this particular setting, that flexibility is enough to support NP-completeness. Also interesting is that both Condorcet and approval, while resistant to
constructive control, are vulnerable to destructive control.
Proof Comments
This section tries to give some general feeling for the proofs.
Complete proofs will be included in the full version of this
paper.
Immunity results are generally clear from the definitions.
For example, note that any system satisfying, as does approval voting, the “unique” version of the Weak Axiom of
Revealed Preference—a unique winner among a collection
of candidates always remains a unique winner among every
subcollection of candidates that includes him/her—is immune to destructive control by deleting candidates, by partition of candidates, and by run-off partition of candidates.
The certifiably-vulnerable results (which here imply the
vulnerable results) range from clear greedy algorithms to
trickier algorithms based on characterizing the ways in
which a candidate can be precluded from winning (or for
the constructive case, made to win). The more surprising
of these have to do with the tie-handling cases of partition
problems—where the chair can at times do shrewd things
(e.g., shift voters counterintuitively to induce ties that kill
off stronger candidates).
For illustration, we now prove that approval voting is
vulnerable/certifiably-vulnerable to destructive control by
partition of voters in the TP model. An easy example shows
that approval voting is susceptible to this type of control: Let
C = {a, b, c}, and define V to consist of the following ten
voters (specified by 0-1 vectors): v1 = v2 = v3 = v4 =
100, v5 = v6 = v7 = 010, and v8 = v9 = v10 = 001.
In (C, V ), a is the unique winner. But if V is partitioned
into V1 = {v1 , v2 , v5 , v6 , v7 } and V2 = V − V1 , b and c
are nominated by the subcommittees (C, V1) and (C, V2), respectively, and tie for winner in the run-off.
Now, given a set of candidates C, a distinguished candidate c ∈ C, and a voter set V , our polynomial-time algorithm for this problem works as follows. If C = {c}, output
“control impossible,” as c must win; else if c already is not
the unique winner, output (V, ∅) as a successful partition.
AAAI-05 / 99
Otherwise, if ||C|| = 2, output “control impossible,” since
in the current case c is the unique winner, so c will win at
least one subcommittee and also the run-off. If none of the
above cases applies, for each a, b ∈ C with ||{a, b, c}|| = 3,
we test whether we can make a strictly beat c in (C, V1) and
make b strictly beat c in (C, V2). For each voter in V , we
focus just on his/her approval of a, b, and c, represented
as the vector abc ∈ {0, 1}3 . Denote the number of voters
with preference 000 or 111 by N , with 001 by Wc , with
110 by Lc , with 100 by Sa , with 010 by Sb , with 101 by
Sac , and with 011 by Sbc . Since c is the approval winner,
Wc + Sbc − (Lc + Sa ) > 0 and Wc + Sac − (Lc + Sb ) > 0.
If Wc − Lc > Sac + Sa + Sbc + Sb − 2 or Sac + Sa = 0
or Sbc + Sb = 0, then this a and b are hopeless, so move on
to consider the next a and b in the loop. Otherwise, we have
Wc − Lc ≤ Sac + Sa + Sbc + Sb − 2 and Sac + Sa > 0
and Sbc + Sb > 0, and output (V1 , V2) as a successful partition, where V1 contains all voters contributing to Sac and Sa ,
and also min(Wc , Sac + Sa − 1) voters contributing to Wc ,
and V2 = V − V1 . In (C, V1), a (strictly) beats c, since a
gets Sac + Sa − min(Wc , Sac + Sa − 1) more Yes votes
than c. And in (C, V2), b (strictly) beats c, since b has
Sbc + Sb + Lc − (Wc − min(Wc , Sac + Sa − 1)) more Yes
votes than c. So, for the construction to work, we must argue
that Sbc + Sb + Lc + min(Wc , Sac + Sa − 1) − Wc > 0. That
is, we need Wc −Lc < min(Wc , Sac +Sa −1)+Sbc +Sb . If
Wc ≤ Sac +Sa −1, this reduces to 0 < Lc +Sbc +Sb , which
follows from the fact that in the current case Sbc + Sb > 0.
And if Wc > Sac + Sa − 1, the desired inequality follows
from the known fact that Wc −Lc ≤ Sac +Sa +Sbc +Sb −2.
To conclude the polynomial algorithm, if in no loop iteration
did we find an a and b that allowed us to output a partition
of voters dethroning c, then output “control impossible.”
Finally, the resistance results are based on clear containments in NP, plus (polynomial-time, many-one) reductions establishing NP-hardness. We whenever possible try
to achieve multiple resistance results via a single proof.
For example, with a single proof we establish all seven
resistance results for plurality voting: destructive control by
adding, deleting, partition (TE and TP), and run-off partition (TE and TP) of candidates,7 and by partition of voters (TP). We now sketch this proof, which is achieved via
one general construction that yields the reductions, each
from the NP-complete problem Hitting Set: Given a set
B = {b1 , b2 , . . . , bm }, a family S = {S1 , S2 , . . . , Sn } of
subsets Si of B, and a positive integer k, does S have a hitting set of size at most k, i.e., is there a set B 0 ⊆ B with
||B 0 || ≤ k such that for each i, Si ∩ B 0 6= ∅? Given a triple
(B, S, k), we construct the following election. The candidate set is C = B ∪ {c, w}. The voter set V is defined as
follows. There are 2(m−k)+2n(k+1)+4 voters of the form
w > c > · · ·, where “· · ·” means that the remaining candidates follow in some arbitrary order. There are 2n(k +1)+5
voters of the form c > w > · · ·. For each i, 1 ≤ i ≤ n, there
7
Our constructions ensure that the distinguished candidate is
never tied for winner in any subelection in the image of the NPhardness reduction. Thus, these results hold both in the TiesEliminate and Ties-Promote models.
are 2(k + 1) voters of the form Si > w > · · ·, where “Si ”
denotes the elements of Si in some arbitrary order. Finally,
for each j, 1 ≤ j ≤ m, there are two voters of the form
bj > c > · · ·.
We show that S has a hitting set of size less than or equal
to k if and only if destructive control by adding candidates
can be executed for the election with qualified candidates
{c, w}, spoiler candidates B, distinguished candidate w, and
voters V . For every candidate d, let sc(d) denote the number
of voters who rank d first. If B 0 is a hitting set of S of size k,
then in the election (B 0 ∪ {c, w}, V ), sc(w) = 2(m − k) +
2n(k + 1) + 4 and sc(c) = 2n(k + 1) + 5 + 2(m − k). It
follows that w is not the unique winner of (B 0 ∪ {c, w}, V ).
For the converse, suppose that w is not the unique winner
of an election (B 0 ∪ {c, w}, V ), where B 0 ⊆ B. We show
that then B 0 is a hitting set of S of size at most k. First
note that for all B 0 ⊆ B and for all b ∈ B 0 , sc(b) < sc(w)
in (B 0 ∪ {c, w}, V ). So, if w is not the unique winner of
(B 0 ∪ {c, w}, V ), then sc(c) ≥ sc(w) in (B 0 ∪ {c, w}, V ).
In (B 0 ∪ {c, w}, V ), sc(c) = 2n(k+1)+5+2(m−||B 0 ||)
and sc(w) = 2(m − k) + 2n(k + 1) + 4 + 2(k + 1)`, where
` is the number of sets in S that are not hit by B 0 (i.e., that
have an empty intersection with B 0 ). Since sc(w) ≤ sc(c),
it follows that 2(m − k) + 2(k + 1)` ≤ 1 + 2(m − ||B 0 ||),
which implies (k + 1)` + ||B 0 || − k ≤ 0. So ` = 0. Thus, B 0
is a hitting set of S of size at most k. It follows that plurality
is resistant to destructive control by adding candidates.
A similar argument can be used to show that S has a hitting set of size at most k if and only if the election with
candidates C, distinguished candidate w, and voters V can
be destructively controlled by deleting at most m − k candidates. (The proof is omitted.) Thus, plurality is resistant to
destructive control by deleting candidates.
Furthermore, S has a hitting set of size at most k if and
only if the election with candidates C, distinguished candidate w, and voters V can be destructively controlled by
partitioning C into C1 = B 0 ∪ {c, w} and C2 = B − B 0 ,
where B 0 is a hitting set of S of size ≤ k. (The proof is omitted. Since w never ties for winner in a subelection, the proof
works for both TE and TP.) A similar argument works for
destructive control for run-off partition of candidates. Thus,
plurality is resistant to destructive control by partition and
run-off partition of candidates (both in TE and TP).
Finally, we show that plurality is resistant to destructive
control by partition of voters in the TP model. To this end,
we reduce from the Hitting Set problem restricted to instances where n(k + 1) + 1 ≤ m − k. (The proof that this
restriction of Hitting Set is still NP-complete is omitted.)
We first show that, in the election (C, V ) constructed
above, for every partition of V into V1 and V2 , w is a
winner of (C, V1) or of (C, V2). For a contradiction, suppose that w is a winner of neither (C, V1) nor (C, V2). Let
x ∈ B ∪ {c} be a winner of (C, V1) with score scV1 (x), and
let y ∈ B ∪ {c} be a winner of (C, V2) with score scV2 (y).
Then scV1 (x) + scV2 (y) ≥ scV (w) + 2. Since w’s score in
(C, V ) is greater than that of any other candidate, x 6= y.
It follows that scV1 (x) + scV2 (y) ≤ scV (c) + scV (bi ) ≤
2n(k +1)+5+2n(k +1)+2 ≤ 2n(k +1)+5+2(m−k) =
scV (w) + 1, a contradiction. We can now show that, assum-
AAAI-05 / 100
ing n(k + 1) + 1 ≤ m − k, S has a hitting set of size at
most k if and only if V can be partitioned such that w is
not the unique winner of (C, V ) in the TP model. If B 0 is a
hitting set of size at most k, let V1 consist of one voter each
of the form c > w > · · · and bj > c > · · ·, for each j,
1 ≤ j ≤ m, and let V2 = V − V1 . Then the winners of
(C, V1) are B 0 ∪ {c}, and the winner of (C, V2) is w. It is
easy to see that c is the unique winner in (B 0 ∪ {c, w}, V ).
For the converse, suppose there is a partition of V such that
w is not the unique winner of the election. However, w is a
winner of a subcommittee’s election, as shown above. It follows that w is not the unique winner of a run-off election involving w, i.e., w is not the unique winner in (D ∪ {w}, V ),
where D ⊆ B ∪ {c}. An argument similar to that in the “destructive control by adding candidates” case shows that S
has a hitting set of size at most k. This completes the proof
sketch.
Conclusions
In this paper, we studied the computational resistance and
vulnerability of three voting systems—plurality, Condorcet,
and approval voting—to destructive control by an election’s
chair in each of seven control scenarios: candidate addition,
suppression, partition, and run-off partition, and voter addition, suppression, and partition. We classified each case as
immune, vulnerable, or computationally resistant. We also
studied the analogous constructive control cases and fully resolved those that were not considered by Bartholdi, Tovey,
and Trick.
We identified cases where a system immune to constructive control still can be vulnerable to destructive control
(e.g., Condorcet voting for control by adding candidates),
and vice versa (e.g., approval voting for control by deleting
candidates). We saw that, among the systems studied, none
is globally superior to the others. Rather, when choosing a
voting system, one’s choice will depend on the types of control against which protection is most desired. Finally, we
saw that—in contrast to some comments in earlier papers—
tie-breaking is a far from minor issue: For those control
types that involve partitions of the candidate or voter set, we
studied two natural tie-handling rules, and we found specific
cases in which the complexity of the corresponding control
problem varies crucially depending on which tie-handling
rule is adopted.
Acknowledgments
We thank Jeroen Snippe for helpful comments, and we thank
Klaus Wagner and his group for hosting a visit during which
this work was done in part.
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